Let's work through the questions from the exercise you uploaded, which involves utility maximization and preferences in economics.

Given Utility Function:

$u(x_1, x_2) = x_1^2 x_2^3$ with prices $p_1$ for $x_1$ and $p_2$ for $x_2$.

(a) Slope of Cornfield's Indifference Curve at the point $(x_1, x_2)$

The slope of an indifference curve is given by the marginal rate of substitution (MRS), which is the ratio of the marginal utilities (MU) of the two goods: $\text{MRS} = \frac{MU_{x_1}}{MU_{x_2}}$

We need to first find the marginal utilities:

• Marginal utility of $x_1$: $MU_{x_1} = \frac{\partial u}{\partial x_1} = 2x_1 x_2^3$

• Marginal utility of $x_2$: $MU_{x_2} = \frac{\partial u}{\partial x_2} = 3x_1^2 x_2^2$

Now, MRS is: $\text{MRS} = \frac{MU_{x_1}}{MU_{x_2}} = \frac{2x_1 x_2^3}{3x_1^2 x_2^2} = \frac{2}{3} \cdot \frac{x_2}{x_1}$

So, the slope of the indifference curve is: $\text{MRS} = \frac{2x_2}{3x_1}$

(b) When the budget line is tangent to the indifference curve

At the tangency point, the slope of the budget line equals the slope of the indifference curve. The slope of the budget line is given by the negative price ratio: $\text{Slope of Budget Line} = -\frac{p_1}{p_2}$

Equating this to the slope of the indifference curve: $\frac{2x_2}{3x_1} = \frac{p_1}{p_2}$

Now solve for $\frac{p_1 x_1}{p_2 x_2}$: $\frac{p_1 x_1}{p_2 x_2} = \frac{3}{2}$

Thus, when Douglas Cornfield is consuming the optimal bundle, the ratio $\frac{p_1 x_1}{p_2 x_2}$ is $\frac{3}{2}$.

The fraction of his income spent on $x_1$ is calculated using the optimal consumption condition: $p_1 x_1 = \frac{3}{5} \cdot \text{Income}$

Thus, Douglas spends 60% of his income on $x_1$.

(c) Utility Function for Other Family Members

Given the family member’s utility function: $U(x_1, x_2) = c x_1^a x_2^b$

The MRS is: $\text{MRS} = \frac{MU_{x_1}}{MU_{x_2}} = \frac{a x_1^{a-1} x_2^b}{b x_1^a x_2^{b-1}} = \frac{a}{b} \cdot \frac{x_2}{x_1}$

At optimal consumption, this equals the price ratio: $\frac{a}{b} \cdot \frac{x_2}{x_1} = \frac{p_1}{p_2}$

Solving for the ratio $\frac{p_1 x_1}{p_2 x_2}$: $\frac{p_1 x_1}{p_2 x_2} = \frac{b}{a}$

The fraction of income spent on $x_1$ will be: $\frac{p_1 x_1}{\text{Income}} = \frac{a}{a + b}$

So, a family member with this utility function spends $\frac{a}{a + b}$ of their income on $x_1$.

Do you need more detailed explanations, or have any specific questions?

Here are 5 follow-up questions based on this information:

1. How does the marginal rate of substitution affect the choice between two goods?
2. What role does the price ratio play in utility maximization?
3. How would an increase in the price of $x_1$ affect the consumption of both $x_1$ and $x_2$?
4. Can a change in income shift the optimal bundle without changing the MRS?
5. What happens if the exponents in the utility function change—how does this affect the consumer’s expenditure pattern?

Tip: The marginal rate of substitution gives insight into how much of one good a consumer is willing to give up for an additional unit of the other, holding utility constant.